Linear Inequalities in Two Variables · 6.1.2
Testing points in an inequality region
Pupils substitute the coordinates of given points into a linear inequality to test whether each point is a solution. By comparing several points, they form a conjecture about which side of the boundary line contains all solutions, then verify this conjecture by testing further points in that region.
The official learning standard (6.1.2)
“Make and verify the conjecture about the points in the region and the solution of certain linear inequalities.”
What it means
Pupils substitute the coordinates of given points into a linear inequality to test whether each point is a solution. By comparing several points, they form a conjecture about which side of the boundary line contains all solutions, then verify this conjecture by testing further points in that region.
How it is examined
Paper 1 may ask whether a specific point satisfies a given inequality. Paper 2 structured questions can require pupils to test several points by substitution, then state and justify a conjecture about the region containing all solutions, usually as a lead-in to shading the region on a graph.
Worked example
Given the inequality 2x + y ≤ 8, test the points A(1, 2), B(3, 5) and C(4, 1) to determine which are solutions of the inequality. Hence, state a conjecture about the region containing the solutions.
- Substitute A(1, 2): 2(1) + 2 = 4. Since 4 ≤ 8 is true, A is a solution.
- Substitute B(3, 5): 2(3) + 5 = 11. Since 11 ≤ 8 is false, B is not a solution.
- Substitute C(4, 1): 2(4) + 1 = 9. Since 9 ≤ 8 is false, C is not a solution.
- Checking against the line 2x + y = 8: A lies below the line, while B and C lie above it.
- Conjecture: all points lying on or below the line 2x + y = 8 satisfy the inequality 2x + y ≤ 8.
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
How do I check if a point satisfies an inequality?
Substitute the point's x and y values into the inequality and simplify. If the resulting statement is true, such as 4 ≤ 8, the point is a solution.
If the statement is false, the point is not a solution.
Why do we test more than one point?
Testing several points reveals a pattern: points on the same side of the boundary line give the same true or false result. This pattern lets you form a conjecture about the whole region, rather than only knowing the result for one single point.
Does a point exactly on the boundary line count as a solution?
It depends on the inequality symbol. If the inequality includes "equal to", using ≤ or ≥, points on the line are solutions.
If it is strict, using < or >, points exactly on the line are not solutions.